I Met a Murderer (1939) [Mark Warrow]: Drowned after falling into the sea while fleeing from police.

Secret Mission (1942) [Raoul de Carnot]: Shot to death by German soldiers while the squad is escaping from their sabotage mission.

Man of Evil (Fanny by Gaslight) (1944) [Lord Manderstoke]: Shot to death in a duel with Stewart Granger.

The Wicked Lady(1945) [Captain Jerry Jackson]: Shot to death by Margaret Lockwood when he confronts her for betraying him. (Thanks to Garrett)

Odd Man Out (Gang War) (1947) [Johnny McQueen]: Shot to death (along with Kathleen Ryan) after being cornered by police, after Kathleen suicidally opens fire to force the police to shoot them.

The Upturned Glass (1947) [Dr. Michael Joyce]: Commits suicide by jumping off of a cliff.

The Reckless Moment (1949) [Martin Donnelly]: Fatally injured in a car crash, he dies shortly afterwards, living just long enough to "confess" and take the blame for a murder committed by Joan Bennett.

One Way Street (1950) [Dr. Frank Matson]: Hit by a car while crossing the street to meet Marta Toren.

The Desert Fox: The Story of Rommel (Rommel, Desert Fox) (1951) [Field Marshal Erwin Rommel]: Commits suicide (off-screen) by taking a cyanide capsule; he is last seen driving away with the two generals who had brought him the capsule. (Thanks to Tony)

Pandora and the Flying Dutchman (1951) [Hendrick van der Zee, the Flying Dutchman]: Plays a ghost who died many centuries ago; his spirit is put to rest when Ava Gardner willingly sacrifices her life for him.

Julius Caesar(1953) [Brutus]: Commits suicide by throwing himself onto his sword, while Edmund Purdom holds it steady. His body is shown again lying in his tent when Marlon Brando comes in to eulogize him.

The Man Between (1953) [Ivo Kern]: Shot by East German police when he creates a distraction to enable Claire Bloom to cross the border.

A Star Is Born (1954) [Norman Maine]: Commits suicide by drowning himself in the ocean. It's been a long time since I've seen this movie, but as I recall, I think we only saw him walking out into the sea, and did not see his body afterwards.

20,000 Leagues Under the Sea (1954) [Captain Nemo]: Killed in an explosion when he deliberately sets off a bomb aboard his submarine, in order to destroy the battleships gathered nearby. (The character was revealed to have survived in the 1961 sequel Mysterious Island, with Herbert Lom taking over the role.) (Thanks to Garrett)

Torpedo Bay(Beta Som; Defi a Gibraltar; Finche dura la tempesta) (1962) [Captain Blayne]: Killed by Gabriele Ferzetti, though the plot synopsis I read did not provide any more specific details.

The Fall of the Roman Empire (1964) [Timonides]: Killed while trying to stop the Roman soldiers from attacking a village.

Genghis Khan(1965) [Kam Ling]: Dragged behind a horse after being tied up in a large sack. (Thanks to Brian)

Cold Sweat (De la part des copains; L'Uomo dalle due ombre) (1970) [Ross]: Shot by Jean Topart during a fight with Charles Bronson; James later dies of his wound despite Liv Ullmann's efforts to save him.

Child's Play(1973) [Jerome Malley]: Commits suicide by jumping off the roof of the school building.

Kill! (Kill! Kill! Kill!) (1972) [Alan Hamilton]: Shot to death in a shoot-out with drug dealers in the jungle; the movie ends with a surreal image (which I suppose was meant to represent James' dying thoughts) of everybody who had been killed in the movie jumping up and down in the middle of the jungle.

Frankenstein: The True Story (1973) [Dr. John Polidori]: Struck by lightning (instantly turning him into a charred skeleton) after Michael Sarrazin ties him to the ship's mast during a storm.

Mandingo (1975) [Warren Maxwell]: Shot to death by one of his slaves when the slave tries to stop James and Perry King from killing Ken Norton. (Thanks to Robert)

The Water Babies(1978; animated/live-action) [Mr. Grimes/Killer Shark]: Playing a dual role in both the live-action and animated sequences, the "Killer Shark" is crushed to death, along with the Electric Eel (voiced by Bernard Cribbins), when the underwater cave/castle collapses. The live-action "Mr. Grimes" survives the movie. (Thanks to ND)

'Salem's Lot(1979) [Richard Straker]: Shot repeatedly in the chest by David Soul. (Thanks to Jesse)

(Thanks to Clive Hirschorn's 1975 book The Films of James Mason for most of the above information.)